For what values of 𝑝 does the sum from 𝑛 equals one to ∞ of one over 𝑛 to the five 𝑝 power converge?
We can, firstly, recognize this series to be a 𝑝-series. Recall that a 𝑝-series is a series of the form the sum from 𝑛 equals one to ∞ of one over 𝑛 to the 𝑝 power. But to avoid confusion with the 𝑝 in the question and the 𝑝 here, let’s replace this 𝑝 with 𝑘. And we have a really useful theorem for 𝑝-series, which says that this 𝑝-series converges if 𝑘 is greater than one and diverges otherwise.
So, it’s the value of the exponent 𝑘 in the denominator, which tells us if the series is convergent. So, for this series to be convergent, five 𝑝 would need to be greater than one. And we can actually divide both sides through by five to get 𝑝 greater than one-fifth.
So, what we’re saying is that the sum from 𝑛 equals one to ∞ of one over 𝑛 raised to the five 𝑝 power converges only if 𝑝 is greater than one-fifth.